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Submitted on 30 Oct 2013
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**Lighting Computations**

### Eric Dumont, Karine Joulan, Vincent Ledoux

**To cite this version:**

CIE 27th Session • Sun City/ZA 1155

**TABULATION AND COMPLETION OF MEASURED BRDF **

**DATA FOR LIGHTING COMPUTATIONS**

**Dumont E., Joulan K., Ledoux V.**
Université Paris-Est, Ifsttar, Paris, France

eric.dumont@ifsttar.fr
**Abstract**

This paper proposes a practical method for the tabulation and completion of measured BRDF data. The BRDF of materials is needed for lighting computations and computer graphics. It can be measured with a gonioreflectometer, but the resulting data is often irregular and always partial. A common solution to get a complete BRDF is to fit the available data with an analytical model. An arguably more accurate solution is to interpolate and extrapolate from the available data. This solution was implemented using Scilab, a free open-source software, and tested on two typical road surface materials: plain asphalt pavement and asphalt pavement painted with retroreflective marking.

Keywords: BRDF, Photometry, Lighting, Luminance, Retroreflection, Grazing Angles
**1 Introduction**

Global illumination techniques have achieved a high degree of physical fidelity which serves many fields of application and particularly lighting design. Some physically-based rendering engines make it possible to account for the actual photometrical properties of light sources, surface materials and even scattering media, without need for analytical models. The problem now lies in the collection of the photometrical data, especially for characterizing reflection properties, since it involves a three-dimensional function (with a fourth dimension for anisotropic materials) whereas emission involves only two dimensions, and scattering usually only needs one.

The reflection properties of a surface material are characterized by means of the luminance coefficient, which is defined as the ratio between the reflected luminance and the illuminance, at any given point on the surface (CIE, 1984). It varies with both the lighting direction and the viewing direction and is thus often referred to as the bidirectional reflectance distribution function (BRDF). It can be measured by means of a gonioreflectometer, with some limits: retroreflection directions are problematic and grazing directions are simply unattainable (Coulomb, 1996).

Once the BRDF data are collected, the problem of implementing them still remains because lighting computations will almost certainly involve directions that were not measured. The common approach consists in approximating the BRDF by fitting analytical or empirical models to the data (Lawrence, 2004). However, direct sampling of the actual data is preferable for accurate photometrical computations (Antonutto, 2004). The reflectance value for a given lighting direction and a given viewing direction can simply be linearly interpolated, provided that the BRDF data is completely tabulated (or “gridded”) – hence the need for a tool to tabulate and complete irregular and partial BRDF data before running lighting computations.

In this paper, we propose a pragmatic solution to perform these operations using Scilab, a free scientific software package which facilitates the manipulation of multi-dimensional arrays (Bunks, 1999). In the Scilab programming environment, it takes little more than 200 lines of code to turn irregular and partial BRDF data into a complete BRDF table.

**2 BRDF measurement and implementation in lighting computations**
**2.1 Definition**

The reflective properties of a surface depend on the material as well as on the texture, especially for
rough surfaces such as those encountered in the road environment. They are characterized by means
*of the luminance coefficient q(w*i*, w*r*), which is defined as the ratio between the reflected luminance L(w*r)

*and the illuminance E(w*i), at any given point on the surface (CIE, 1984). It varies with both the lighting

*direction w*i* and the viewing direction w*r as illustrated in Figure 1. It is thus often referred to as the

Bidirectional Reflectance Distribution Function (BRDF).

(1) A Lambertian, or diffuse, surface reflects a constant luminance in every directions however it is illuminated. A specular, or mirror-like, surface reflects light only in the direction symmetrical to the lighting direction with respect to the surface normal (following Snell-Descartes law). Real-world surfaces generally fall between these theoretical cases. The road environment also contains retroreflective materials, designed to reflect light back where it came from in order to optimize vehicle lighting.

From this point on, we only consider isotropic surface materials. The BRDF is then independent of the
*lighting half-plane, and may be described with three angles: q*i*, j*r* and q*r* (with j*i set to zero). It should

be noted that this consideration is reasonable for most but not all materials in the road environment: for instance, microprism sheetings on road signs are definitely anisotropic.

*q*

_{i}

**n**

**n**

### w

_{i}

### w

_{r}

*q*

_{r}

*f*

_{r}

*f*

_{i}

**Figure 1. The BRDF is generally a function of 4 angles**
**2.2 Properties**

The BRDF obeys two physical principles: energy conservation and Helmholtz reciprocity. The energy
conservation property states that a surface cannot reflect more light than it receives. It yields the following
*inequality on the directional hemispherical reflectance r(q*_{i}):

(2) Helmholtz reciprocity principle states that the luminance coefficient stays the same when the lighting and viewing directions are interchanged. Combined with the symmetry of the BRDF about the lighting plane, it yields the following equality:

(3)
**2.3 Measurement**

The BRDF is measured by means of a device called a gonioreflectometer, as it measures light
reflected from an illuminated surface sample at different angles. For a thorough characterization, a
gonioreflectometer ought to cover a hemisphere over the sample. However, there are geometrical
constraints which will prevent measurement in certain directions or combinations of directions, namely
*retroreflection (q*_{r} = *q*_{i}* ; j*_{r} = 0) and grazing angles (q_{i} ≈ p/2 and q_{r} ≈ p/2, "j_{ r}).

*L*(ω_{ω}_{ω}_{ω}_{r}*) and the illuminance E(ω*_{ω}_{ω}_{ω}_{i}), at any given point on the surface (CIE, 1984). It varies with both the
lighting direction _{ω}_{ω}_{ω}_{ω}_{i} and the viewing direction ω_{ω}_{ω}_{ω}_{r} as illustrated in Figure 1. It is thus often referred to as
the Bidirectional Reflectance Distribution Function (BRDF).

### (

### )

### (

### )

### ( )

i r r i _{d}d

_{}

*E*

*L*

*,*

*q*

_{=}

_{(1) }

A Lambertian, or diffuse, surface reflects a constant luminance in every directions however it is illuminated. A specular, or mirror-like, surface reflects light only in the direction symmetrical to the lighting direction with respect to the surface normal (following Snell-Descartes law). Real-world surfaces generally fall between these theoretical cases. The road environment also contains retroreflective materials, designed to reflect light back where it came from in order to optimize vehicle lighting.

From this point on, we only consider isotropic surface materials. The BRDF is then independent of the
lighting half-plane, and may be described with three angles: _{θ}_{i}, _{ϕ}_{r} and _{θ}_{r} (with _{ϕ}_{i} set to zero). It should
be noted that this consideration is reasonable for most but not all materials in the road environment:
for instance, microprism sheetings on road signs are definitely anisotropic.

### θ

_{i}

**n**

**n**

### ω

### ω

### ω

### ω

_{i}

### ω

### ω

### ω

### ω

r### θ

r### φ

r### φ

i**Figure 1. The BRDF is generally a function of 4 angles **
**2.2 Properties **

The BRDF obeys two physical principles: energy conservation and Helmholtz reciprocity. The energy
conservation property states that a surface cannot reflect more light than it receives. It yields the
following inequality on the directional hemispherical reflectance _{ρ}(_{θ}_{i}):

### ( )

θi =###

### (

θi,ϕr,θr### )

cosθrdωr ≤1ρ *q* _{(2) }

Helmholtz reciprocity principle states that the luminance coefficient stays the same when the lighting and viewing directions are interchanged. Combined with the symmetry of the BRDF about the lighting plane, it yields the following equality:

### (

θi,ϕr,θr### )

*q*

### (

θr,2 ϕr,θi### )

*q*

### (

θr,ϕr,θi### )

*q* _{=} _{−} _{=} _{(3) }

**2.3 Measurement **

The BRDF is measured by means of a device called a gonioreflectometer, as it measures light reflected from an illuminated surface sample at different angles. For a thorough characterization, a gonioreflectometer ought to cover a hemisphere over the sample. However, there are geometrical constraints which will prevent measurement in certain directions or combinations of directions, namely retroreflection (θr = θi ; ϕr = 0) and grazing angles (θi ≈ π/2 and θr ≈ π/2, ∀ϕ r).

Ifsttar’s photometry laboratory owns a gonioreflectometer that was designed especially for measuring
*L*(ω_{ω}_{ω}_{ω}_{r}*) and the illuminance E(ω*_{ω}_{ω}_{ω}_{i}), at any given point on the surface (CIE, 1984). It varies with both the
lighting direction _{ω}_{ω}_{ω}_{ω}_{i} and the viewing direction _{ω}_{ω}_{ω}_{ω}_{r} as illustrated in Figure 1. It is thus often referred to as
the Bidirectional Reflectance Distribution Function (BRDF).

### (

### )

### (

### )

### ( )

i r r i _{d}d

_{}

*E*

*L*

*,*

*q*

_{=}

_{(1) }

A Lambertian, or diffuse, surface reflects a constant luminance in every directions however it is illuminated. A specular, or mirror-like, surface reflects light only in the direction symmetrical to the lighting direction with respect to the surface normal (following Snell-Descartes law). Real-world surfaces generally fall between these theoretical cases. The road environment also contains retroreflective materials, designed to reflect light back where it came from in order to optimize vehicle lighting.

From this point on, we only consider isotropic surface materials. The BRDF is then independent of the lighting half-plane, and may be described with three angles: θi, ϕr and θr (with ϕi set to zero). It should

be noted that this consideration is reasonable for most but not all materials in the road environment: for instance, microprism sheetings on road signs are definitely anisotropic.

### θ

i**n**

**n**

### ω

### ω

### ω

### ω

i### ω

### ω

### ω

### ω

r### θ

r### φ

r### φ

i**Figure 1. The BRDF is generally a function of 4 angles **
**2.2 Properties **

The BRDF obeys two physical principles: energy conservation and Helmholtz reciprocity. The energy
conservation property states that a surface cannot reflect more light than it receives. It yields the
following inequality on the directional hemispherical reflectance _{ρ}(_{θ}_{i}):

### ( )

θi =###

### (

θi,ϕr,θr### )

cosθrdωr ≤1ρ *q* _{(2) }

Helmholtz reciprocity principle states that the luminance coefficient stays the same when the lighting and viewing directions are interchanged. Combined with the symmetry of the BRDF about the lighting plane, it yields the following equality:

### (

θi,ϕr,θr### )

*q*

### (

θr,2 ϕr,θi### )

*q*

### (

θr,ϕr,θi### )

*q* _{=} _{−} _{=} _{(3) }

**2.3 Measurement **

The BRDF is measured by means of a device called a gonioreflectometer, as it measures light
reflected from an illuminated surface sample at different angles. For a thorough characterization, a
gonioreflectometer ought to cover a hemisphere over the sample. However, there are geometrical
constraints which will prevent measurement in certain directions or combinations of directions, namely
retroreflection (_{θ}_{r}_{ = θ}_{i} ; _{ϕ}_{r} = 0) and grazing angles (θ_{i} ≈ π/2 and θ_{r} ≈ π/2, ∀ϕ_{ r}).

Ifsttar’s photometry laboratory owns a gonioreflectometer that was designed especially for measuring
the BRDF of rough surface materials such as road pavement, with a measurement area of 10 cm in
*L*(ω_{ω}_{ω}_{ω}_{r}*) and the illuminance E(ω*_{ω}_{ω}_{ω}_{i}), at any given point on the surface (CIE, 1984). It varies with both the

lighting direction _{ω}_{ω}_{ω}_{ω}_{i} and the viewing direction _{ω}_{ω}_{ω}_{ω}_{r} as illustrated in Figure 1. It is thus often referred to as
the Bidirectional Reflectance Distribution Function (BRDF).

### (

### )

### (

### )

### ( )

i r r i _{d}d

_{}

*E*

*L*

*,*

*q*

_{=}

_{(1) }

A Lambertian, or diffuse, surface reflects a constant luminance in every directions however it is illuminated. A specular, or mirror-like, surface reflects light only in the direction symmetrical to the lighting direction with respect to the surface normal (following Snell-Descartes law). Real-world surfaces generally fall between these theoretical cases. The road environment also contains retroreflective materials, designed to reflect light back where it came from in order to optimize vehicle lighting.

From this point on, we only consider isotropic surface materials. The BRDF is then independent of the
lighting half-plane, and may be described with three angles: _{θ}_{i}, _{ϕ}_{r} and _{θ}_{r} (with _{ϕ}_{i} set to zero). It should
be noted that this consideration is reasonable for most but not all materials in the road environment:
for instance, microprism sheetings on road signs are definitely anisotropic.

### θ

i**n**

**n**

### ω

### ω

### ω

### ω

i### ω

### ω

### ω

### ω

_{r}

### θ

r### φ

r### φ

_{i}

**Figure 1. The BRDF is generally a function of 4 angles **
**2.2 Properties **

The BRDF obeys two physical principles: energy conservation and Helmholtz reciprocity. The energy conservation property states that a surface cannot reflect more light than it receives. It yields the following inequality on the directional hemispherical reflectance ρ(θi):

### ( )

θi =###

### (

θi,ϕr,θr### )

cosθrdωr ≤1ρ *q* _{(2) }

Helmholtz reciprocity principle states that the luminance coefficient stays the same when the lighting and viewing directions are interchanged. Combined with the symmetry of the BRDF about the lighting plane, it yields the following equality:

### (

θi,ϕr,θr### )

*q*

### (

θr,2 ϕr,θi### )

*q*

### (

θr,ϕr,θi### )

*q* _{=} _{−} _{=} _{(3) }

**2.3 Measurement **

The BRDF is measured by means of a device called a gonioreflectometer, as it measures light
reflected from an illuminated surface sample at different angles. For a thorough characterization, a
gonioreflectometer ought to cover a hemisphere over the sample. However, there are geometrical
constraints which will prevent measurement in certain directions or combinations of directions, namely
retroreflection (_{θ}_{r}_{ = θ}_{i} ; _{ϕ}_{r} = 0) and grazing angles (θ_{i} ≈ π/2 and θ_{r} ≈ π/2, ∀ϕ_{ r}).

CIE 27th Session • Sun City/ZA 1157 Ifsttar’s photometry laboratory owns a gonioreflectometer that was designed especially for measuring the BRDF of rough surface materials such as road pavement, with a measurement area of 10 cm in diameter (Coulomb, 1996). This gonioreflectometer, illustrated in Figure 2, was also designed for grazing lighting angles (up to 85°) and even more grazing viewing angles (up to 89°) with drivers’ experience in mind. It also closely approaches retroreflection (within 2,5°). Within these limits, any list of combinations of angles may be input for measurement by the device. Results are saved to an ASCII file with a single comment line followed by 4-column data lines containing the values of the 3 angles and the luminance coefficient for each combination.

source
sensor
sample
source
sensor
sample _{q}_{i}
*q*r
*j*r
Li
ght
ing
dir
ec
tio
n
Vi
ew
ing
dir
ec
tio
n

**Figure 2. Ifsttar’s 3-angle gonioreflectometer was designed **
to measure the BRDF of road surface samples

**2.4 Implementation in lighting computations**

The point of measuring the BRDF of a material is to calculate its luminance under various illumination and observation conditions. However, because of the geometric limits at retroreflection and grazing angles, because of possible measurement errors (due to under- or over-exposure), and simply because not all angle combinations are measured, some luminance coefficient values are unavailable in the collected BRDF data. Hence, a method is needed to determine the luminance coefficient for angles that have not been measured.

The common solution consists in approximating the BRDF by fitting analytical (theoretical or empirical)
models to the data (Lawrence, 2004). However, direct sampling of the actual data is arguably preferable
for accurate photometrical computations (Antonutto, 2004), especially for rough road surface materials.
The reflectance value for a given lighting direction and a given viewing direction can simply be linearly
interpolated, provided that the BRDF data is completely tabulated (or “gridded”). Hence the need for
a tool to tabulate and complete irregular and partial BRDF data before running lighting computations.
**3 Tabulation and completion of measured BRDF data**

**3.1 Using Scilab**

**2.1 Tabulation**

Tabulation is the first step. It involves browsing the data to extract the number and values of the angles
*q*i*, j*r* and q*r at which the measurement were performed. It is possible to increase the size of the array by

implementing the Helmholtz reciprocity principle (Equation 3), which will provide reflectance values at
angles which were not actually measured. In case of duplicate combinations, the reflectance values can
*simply be averaged. The data are then sorted by ascending angle value, first with q*i*, then with j*r and

*finally with q*r. At this point the reflectance values are still stored in the 4th column of a 4-column matrix.

**2.1 Completion**

The next step consists in interpolating the BRDF values in the retroreflection directions. It can be
*achieved by a call to the bidimensional cubic Shepard interpolation function for each lighting angle q*i.

This is where the data are “gridded” using Scilab hypermatrices. The choice of the Shepard function was dictated by the irregular nature of the data.

*The final operation consists in extrapolating BRDF values at q*i = 90°, the perfectly grazing lighting angle.

It can be achieved by a call to the cubic spline interpolation function. Contrary to the Shepard function, the spline function allows extrapolation, but only in one dimension for irregular data. The BRDF at the perfectly grazing viewing angle does not need to be extrapolated, because lighting computations involve

*q.cos(q*r*), which is null at q*r = 90°.

The result is a 3D-array which can be saved in an adapted version of the Radiant Imaging BSDF data interchange file format, arbitrarily adopted for lack of a standard format. It involves calculating the directional hemispherical reflectance expressed in Equation 2. The previous operations were programmed into a Scilab script. This script contains less than 250 lines, including input/output instructions and comments.

**4 Sample results**

The proposed method for tabulating and completing measured BDRF data was tested on two samples of asphalt pavement, one painted with retro-reflective painting. The samples are presented in Figure 3.

(a) Plain pavement (b) Pavement with retroreflective paint
**Figure 3. Pictures of the sample road surfaces on which the BRDF **

was measured, and then tabulated and completed

Both rough surface samples were measured with Ifsttar’s 3-angle goniophotometer, presented in Section
*2.3. The lighting angle q*i* was sampled every 10° between 0° and 80°. The viewing angle q*r was sample

every 5° between 0° and 85°, plus 88° and 89°. The angle between the lighting and viewing half-planes
*j*_{r} was sampled every 10° between 0° and 180°, plus 2° and 5°. Raw and completed BRDF data are
presented for several lighting angles in Figure 4 for the plain pavement sample and Figure 5 for the
painted pavement sample. Two obvious observations can be made: the plain pavement surface is dark
and shows specularity with a little retroreflectivity at high lighting angles, while the painted pavement
surface is light and retroreflective, especially at high lighting angles.

CIE 27th Session • Sun City/ZA 1159 maps are presented in Figure 6. It appears that the slight retroreflectivity of the pavement makes quite a difference.

**Plain pavement (raw)**

0,01 0,1 1 10

-90 -60 -30 0 30 60 90

**Viewing angle (<0 for retroreflection)**

**Lu**
**m**
**in**
**an**
**ce co**
**ef**
**fici**
**en**
**t**
**(cd**
**.m**
**-2.**
**lx-1)**
0
20
40
60
80

**Plain pavement (completed)**

0,01 0,1 1 10

-90 -60 -30 0 30 60 90

**Viewing angle (<0 for retroreflection)**

**Lu**
**m**
**in**
**an**
**ce co**
**ef**
**fici**
**en**
**t**
**(cd**
**.m**
**-2.**
**lx-1)** _{0}
20
40
60
80
90

**Painted pavement (raw)**
0,01
0,1
1
10
-90 -60 -30 0 30 60 90

**Viewing angle (<0 for retroreflection)**

**Lu**
**m**
**in**
**an**
**ce co**
**ef**
**fici**
**en**
**t**
**(cd**
**.m**
**-2.**
**lx-1)**
0
20
40
60
80

**Painted pavement (completed)**

0,01 0,1 1 10

-90 -60 -30 0 30 60 90

**Viewing angle (<0 for retroreflection)**

**Lu**
**m**
**in**
**an**
**ce co**
**ef**
**fici**
**en**
**t**
**(cd**
**.m**
**-2.**
**lx-1)** _{0}
20
40
60
80
90

CIE 27th Session • Sun City/ZA 1161
**Figure 6. Luminance map of a road scene illuminated by headlamps simulated with PROF-LCPC **

with analytical BRDF models (top) and measured BRDF data (bottom)
**5 Discussion**

A practical method to tabulate and complete BRDF data measured with a 3-angle gonioreflectometer has been proposed. The resulting data facilitates lighting computations based on interpolation into the photometric data, which is arguably more accurate than using analytical models.

It should however be emphasized that the interpolation and extrapolation operations performed for the completion will result in sound data if the combinations of angles at which the measurements are made are chosen soundly. It is recommended to sample more densely at specular and retroreflection directions, and to go as far as possible toward grazing directions.

Just like PROF-LCPC software, the Scilab script will be distributed freely to anyone who sends a demand by e-mail to the first author of the present paper.

**Plain pavement**
-50%
-25%
0%
25%
50%
75%
100%
0,5° 1,0° 1,5° 2,0° 2,5°

**Difference between lighting and viewing angle**

**R**
**el**
**at**
**ive er**
**ro**
**r o**
**n**
**lu**
**m**
**in**
**an**
**ce co**
**ef**
**fici**
**en**
**t**
60
70
80
85
**Painted pavement**
-50%
-25%
0%
25%
50%
75%
100%
0,5° 1,0° 1,5° 2,0° 2,5°

**Difference between lighting and viewing angle**

**R**
**el**
**at**
**ive er**
**ro**
**r o**
**n**
**lu**
**m**
**in**
**an**
**ce co**
**ef**
**fici**
**en**
**t**
60
70
80
85

**Figure 7. Comparison between measured and completed BRDF data for the plain pavement sample **
(top) and the painted pavement sample (bottom) at different lighting angles.

**Acknowledgements**

This work is part of the French project eMotive, sponsored by the Ministry of Economy, Finance and Industry within the FUI framework.

**References**

ANTONUTTO, G. 2004. Road Lighting Simulation in Radiance. 3rd International Radiance Workshop, Fribourg, October 11-12.

BUNKS, C., CHANCELIER, J.P., DELEBECQUE, F., GOMEZ, C., GOURSAT, M., NIKOUKHAH, R.,
*STEER, S. 1999. Engineering and Scientific Computing with Scilab. Boston: Birkhaüser.*

*CIE 1984. CIE 066:1984. Road Surfaces and Lighting. Vienna: CIE.*

CIE 27th Session • Sun City/ZA 1163
*DUMONT, E. 1999. Semi-Monte Carlo Light Tracing for the Study of Road Visibility in Fog. In Monte *
*Carlo and Quasi-Monte Carlo Methods 1998. Berlin: Springer.*